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Simplifying Complex NumbersDYNAMICAL SYSTEMSDynamical systems is a branch of mathematics that studies

constantly changing systems like the stock market, the weather, and population. In many cases, one can catch a glimpse of the system at some point in time, but the forces that act on the system cause it to change quickly. By analyzing how a dynamical system changes over time, it may be possible to predict the behavior of the system in the future. Oneof the basic mathematical models of a dynamical system is iteration of a complexfunction. A problem related to this will be solved in Example 4.

Recall that complex numbers are numbers of the form a � bi, where a and bare real numbers and i, the imaginary unit, is defined by i2 � �1. The first fewpowers of i are shown below.

Notice the repeating pattern of the powers of i.

i, �1, �i, 1, i, �1, �i, 1

In general, the value of in, where n is a wholenumber, can be found by dividing n by 4 andexamining the remainder as summarized in the table at the right.

You can also simplify any integral power of i byrewriting the exponent as a multiple of 4 plus apositive remainder.

Simplify each power of i.

a. i53 b. i�13

Method 1 Method 2 Method 1 Method 253 � 4 � 13 R1 i53 � (i4)13 � i �13 � 4 � �4 R3 i�13 � (i4)�4 � i3

If R � 1, in � i. � (1)13 � i If R � 3, in � �i. � (1)�4 � i3

i53 � i � i i�13 � �i � �i

580 Chapter 9 Polar Coordinates and Complex Numbers

9-5

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OBJECTIVE• Add, subtract,

multiply, anddivide complexnumbers inrectangularform.

5/25 8/25

10,466

10,896

11,326

A Typical Graph of the Stock Market

i1 � i i2 � �1 i3 � i2 � i � �i i4 � (i2)2 � 1

i5 � i4 � i � i i6 � i4 � i2 � �1 i7 � i4 � i3 � �i i8 � (i2)4 � 1

To find the value of in, let R be the remainder

when n is divided by 4.

if R � 0 in � 1if R � 1 in � iif R � 2 in � �1if R � 3 in � �i

Example 1

�

The complex number a � bi, where a and b are real numbers, is said to be inrectangular form. a is called the real part and b is called the imaginary part. If b � 0, the complex number is a real number. If b � 0, the complex number is animaginary number. If a � 0 and b � 0, as in 4i, then the complex number is apure imaginary number. Complex numbers can be added and subtracted byperforming the chosen operation on both the real and imaginary parts.

Simplify each expression.

a. (5 � 3i) � (�2 � 4i)

(5 � 3i) � (�2 � 4i) � [5 � (�2)] � [�3i � 4i]

� 3 � i

b. (10 � 2i) � (14 � 6i)

(10 � 2i) � (14 � 6i) � 10 � 2i � 14 � 6i

� �4 � 4i

The product of two or more complex numbers can be found using the sameprocedures you use when multiplying binomials.

Simplify (2 � 3i)(7 � 4i).

(2 � 3i)(7 � 4i) � 7(2 � 3i) � 4i(2 � 3i) Distributive property� 14 � 21i � 8i � 12i2 Distributive property� 14 � 21i � 8i � 12(�1) i2 � �1� 2 � 29i

Iteration is the process of repeatedly applying a function to the outputproduced by the previous input. When using complex numbers with functions, it is traditional to use z for the independent variable.

DYNAMICAL SYSTEMS If f(z) � (0.5 � 0.5i)z, find the first five iterates of f for the initial value z0 � 1 � i. Describe any pattern that you see.

f(z) � (0.5 � 0.5i)z

f(1 � i) � (0.5 � 0.5i)(1 � i) Replace z with 1 � i.� 0.5 � 0.5i � 0.5i � 0.5i2

� i z1 � i

f(i) � (0.5 � 0.5i)i� 0.5i � 0.5i2

� �0.5 � 0.5i z2 � �0.5 � 0.5i

f(�0.5 � 0.5i) � (0.5 � 0.5i)(�0.5 � 0.5i)� �0.25 � 0.25i � 0.25i � 0.25i2

� �0.5 z3 � �0.5

(continued on the next page)

Lesson 9-5 Simplifying Complex Numbers 581

GraphingCalculatorTip

Some calculators have acomplex number mode.In this mode, they canperform complexnumber arithmetic.

Example 2

Example 3

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Example 4

�f(�0.5) � (0.5 � 0.5i )(�0.5)

� �0.25 � 0.25i z4 � �0.25 � 0.25i

f(�0.25 � 0.25i ) � (0.5 � 0.5i )(�0.25 � 0.25i )

� �0.125 � 0.125i � 0.125i � 0.125i2

� �0.25i z5 � �0.25i

The first five iterates of 1 � i are i, �0.5 � 0.5i, �0.5, �0.25 � 0.25i, and�0.25i. The absolute values of the nonzero real and imaginary parts (1, 0.5,0.25) stay the same for two steps and then are halved.

Two complex numbers of the form a � bi and a � bi are called complexconjugates. Recall that if a quadratic equation with real coefficients has complexsolutions, then those solutions are complex conjugates. Complex conjugates alsoplay a useful role in the division of complex numbers. To simplify the quotient of two complex numbers, multiply the numerator and denominator by theconjugate of the denominator. The process is similar to rationalizing the

denominator in an expression like .

Simplify (5 � 3i) (1 � 2i).

(5 � 3i) � (1 � 2i ) � �51

�

�

32ii

�

� �51

�

�

32ii

� � �11

�

�

22ii

�

�

� i2 � �1

� �11 �

57i

�

� �151� � �

75

� i Write the answer in the form a � bi.

The list below summarizes the operations with complex numbers presentedin this lesson.

5 � 7i � 6(�1)��

1 � (�4)

5 � 10i � 3i � 6i 2��

1 � 4i 2

1�3 � �2�

GraphingCalculatorProgramsTo download agraphingcalculatorprogram thatperformscomplexiteration, visit:www.amc.glencoe.com

Multiply by 1; 1 � 2i is theconjugate of 1 � 2i.

Example 5

For any complex numbers a � bi and c � di, the following are true.

(a � bi ) � (c � di ) � (a � c) � (b � d )i

(a � bi ) � (c � di ) � (a � c) � (b � d )i

(a � bi )(c � di ) � (ac � bd ) � (ad � bc)i

�ac �

�

dbii

� � �cac

2 �

�

dbd

2� � �

cbc

2 �

�

dad

2� i

Operationswith Complex

Numbers


http://www.amc.glencoe.com

CommunicatingMathematics

Guided Practice

Practice

Read and study the lesson to answer each question.

1. Describe how to simplify any integral power of i.

2. Draw a Venn diagram to show the relationship between real, pure imaginary,and complex numbers.

3. Explain why it is useful to multiply by the conjugate of the denominator overitself when simplifying a fraction containing complex numbers.

4. Write a quadratic equation that has two complex conjugate solutions.

Simplify.

5. i�6 6. i10 � i2 7. (2 � 3i) � (�6 � i)

8. (2.3 � 4.1i) � (�1.2 � 6.3i) 9. (2 � 4i) � (�1 � 5i)

10. (�2 � i)2 11. �1 �

i2i

�

12. Vectors It is sometimes convenient to use complex numbers to representvectors. A vector with a horizontal component of magnitude a and a verticalcomponent of magnitude b can be represented by the complex number a � bi.If an object experiences a force with a horizontal component of 2.5 N and avertical component of 3.1 N as well as a second force with a horizontalcomponent of �6.2 N and a vertical component of 4.3 N, find the resultant forceon the object. Write your answer as a complex number.

Simplify.

13. i6 14. i19 15. i1776 16. i9 � i�5

17. (3 � 2i) � (�4 � 6i) 18. (7 � 4i) � (2 � 3i)

19. ��12

� � i� � (2 � i) 20. (�3 � i) � (4 � 5i)

21. (2 � i)(4 � 3i) 22. (1 � 4i)2

23. �1 � �7�i��2 � �5�i� 24. �2 � ��3��1 � ��12��25. �

12

�

�

2ii

� 26. ��

34�

�

2ii

� 27. �55

�

�

ii

�

28. Write a quadratic equation with solutions i and �i.

29. Write a quadratic equation with solutions 2 � i and 2 � i.

Simplify.

30. (2 � i)(3 � 2i)(1 � 4i) 31. (�1 � 3i)(2 � 2i)(1 � 2i)

32. 33.

34. �(23

�

�

ii)2� 35. �(�

(13

�

�

i2)i

2

)2�

2 � �2�i��3 � �6�i

�12

� � �3�i��1 � �2�i


C HECK FOR UNDERSTANDING

Look BackYou can refer toLessons 8-1 and 8-2to review vectors,components, andresultants.

E XERCISES

A

B

C

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Applicationsand ProblemSolving

Mixed Review

36. Electricity Impedance is a measure of how much hindrance there is to the flowof charge in a circuit with alternating current. The impedance Z depends on theresistance R, the reactance due to capacitance XC, and the reactance due toinductance XL in the circuit. The impedance is written as the complex number Z � R � (XL � XC)j. (Electrical engineers use j to denote the imaginary unit.) In the first part of a particular series circuit, the resistance is 10 ohms, thereactance due to capacitance is 2 ohms, and the reactance due to inductance is 1 ohm. In the second part of the circuit, the respective values are 3 ohms, 1 ohm, and 1 ohm.

a. Write complex numbers that represent the impedances in thetwo parts of the circuit.

b. Add your answers from part a to find the total impedance inthe circuit.

c. The admittance of an AC circuit is a measure of how well thecircuit allows current to flow. Admittance is the reciprocal

of impedance. That is, S � �Z1

�. The units for admittance are siemens. Find the admittance in a circuit with an impedance of6 � 3j ohms.

37. Critical Thinkinga. Solve the equation x2 � 8ix � 25 � 0.b. Are the solutions complex conjugates?c. How does your result in part b compare with what you already know about

complex solutions to quadratic equations?d. Check your solutions.

38. Critical Thinking Sometimes it is useful to separate a complex function into its real and imaginary parts. Substitute z � x � yi into the function f(z) � z2 to write the equation of the function in terms of x and y only. Simplifyyour answer.

39. Dynamical Systems Find the first five iterates for the given function and initialvalue.a. f(z) � iz, z0 � 2 � ib. f(z) � (0.5 � 0.866i)z, z0 � 1 � 0i

40. Critical Thinking Simplify (1 � 2i)�3.

41. Physics One way to derive the equation of motion in a spring-mass system isto solve a differential equation. The solutions of such a differential equationtypically involve expressions of the form cos �t � i sin �t. You generally expectsolutions that are real numbers in such a situation, so you must use algebra toeliminate the imaginary numbers. Find a relationship between the constants c1and c2 such that c1(cos 2t � i sin 2t) � c2(cos 2t � i sin 2t) is a real number forall values of t.

42. Write the equation 6x � 2y � �3 in polar form. (Lesson 9-4)

43. Graph the polar equation r � 4�. (Lesson 9-2)

44. Write a vector equation of the line that passes through P(�3, 6) and is parallelto v�1, �4. (Lesson 8-6)


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45. Find an ordered triple to represent u� if u� � �14

�v� � 2w�, v� �� 8, 6, 4�, and w� � �2, �6, 3�. (Lesson 8-3)

46. If � and � are measures of two first quadrant angles, find cos (� � �) if

tan � � �43

� and cot � � �152�. (Lesson 7-3)

47. A twig floats on the water, bobbing up and down. The distance between itshighest and lowest points is 7 centimeters. It moves from its highest pointdown to its lowest point and back up to its highest point every 12 seconds.Write a cosine function that models the movement of the twig in relationshipto the equilibrium point. (Lesson 6-6)

48. Surveying A surveyor finds that the angle of elevation from a certain point to the top of a cliff is 60°. From a point 45 feet farther away, the angle of elevation to the top of the cliff is 52°. How high is the cliff to the nearest foot?(Lesson 5-4)

49. What type of polynomial function would be the best modelfor the set of data? (Lesson 4-8)

50. Construction A community wants to build a second pool at their communitypark. Their original pool has a width 5 times its depth and a length 10 timesits depth. They wish to make the second pool larger by increasing the widthof the original pool by 4 feet, increasing the length by 6 feet, and increasingthe depth by 2 feet. The volume of the new pool will be 3420 cubic feet. Findthe dimensions of the original pool. (Lesson 4-4)

51. If y varies jointly as x and z and y � 80 when x � 5 and z � 8, find y when x � 16 and z � 2. (Lesson 3-8)

52. If f(x) � 7 � x2, find f �1(x). (Lesson 3-4)

53. Find the maximum and minimum values of the function f(x, y) � �2x � y for the polygonal convex set determined by the system of inequalities.(Lesson 2-6)x � 6y 1y � x � 2

54. Solve the system of equations. (Lesson 2-2)x � 2y � 7z � 14�x � 3y � 5z � �215x � y � 2z � �7

55. SAT/ACT Practice If BC � BD in the figure, what is the value of x � 40?

A 100 B 80 C 60 D 40

E cannot be determined from the informationgiven


x �3 �1 1 3 5 7 9 11f(x) �4 �2 3 8 6 1 �3 �8

x˚

120˚A CB

D

Extra Practice See p. A43.

The Complex Plane and PolarForm of Complex Numbers

FRACTALS One of the standard ways to generate a fractal involvesiteration of a quadratic function. If the function f(z) � z2 is iteratedusing a complex number as the initial input, there are three possible

outcomes. The terms of the sequence of outputs, called the orbit, may

• increase in absolute value,• decrease toward 0 in absolute value, or• always have an absolute value of 1.

One way to analyze the behavior of the orbit is to graph the numbers in the complexplane. Plot the first five members of the orbit of z0 � 0.9 � 0.3i under iteration byf(z) � z2. This problem will be solved in Example 3.

Recall that a � bi is referred to as the rectangular form of a complex number.The rectangular form is sometimes written as an ordered pair, (a, b). Twocomplex numbers in rectangular form are equal if and only if their real parts areequal and their imaginary parts are equal.

Solve the equation 2x � y � 3i � 9 � xi � yi for x and y, where x and y arereal numbers.

2x � y � 3i � 9 � xi � yi(2x � y) � 3i � 9 � (x � y)i On each side of the equation, group the real

parts and the imaginary parts.

2x � y � 9 and x � y � 3 Set the corresponding parts equal to each other.x � 4 and y � 1 Solve the system of equations.

Complex numbers can be graphed in the complex plane. The complex plane has a real axis andan imaginary axis. The real axis is horizontal, and theimaginary axis is vertical. The complex number a � biis graphed as the ordered pair (a, b) in the complexplane. The complex plane is sometimes called theArgand plane.

Recall that the absolute value of a real number is itsdistance from zero on the number line. Similarly, theabsolute value of a complex number is its distance fromzero in the complex plane. When a � bi is graphed in thecomplex plane, the distance from zero can be calculatedusing the Pythagorean Theorem.


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OBJECTIVES• Graph complex

numbers in thecomplex plane.

• Convert complexnumbers fromrectangular topolar form andvice versa.

Example 1

imaginary (i )

real (�)O

i a � bi

b

a �O

Graph each number in the complex plane and find its absolute value.

a. z � 3 � 2i b. z � 4i

z � 3 � 2i z � 0 � 4i

z � �32 � 2�2� z � �02 � 4�2�� 13� � 4

FRACTALS Refer to the application at the beginning of the lesson. Plot the first five members of the orbit of z0 � 0.9 � 0.3i under iteration by f(z) � z2.

First, calculate the first five members of the orbit. Round the real andimaginary parts to the nearest hundredth.

z1 � 0.72 � 0.54i z1 � f(z0)

z2 � 0.23 � 0.78i z2 � f(z1)

z3 � �0.55 � 0.35i z3 � f(z2)

z4 � 0.18 � 0.39i z4 � f(z3)

z5 � �0.12 � 0.14i z5 � f(z4)

Then graph the numbers in the complex plane.The iterates approach the origin, so their absolutevalues decrease toward 0.

So far we have associated the complex number a � biwith the rectangular coordinates (a, b). You know fromLesson 9-1 that there are also polar coordinates (r, �)associated with the same point. In the case of a complexnumber, r represents the absolute value, or modulus, of thecomplex number. The angle � is called the amplitude orargument of the complex number. Since � is not unique, itmay be replaced by � � 2�k, where k is any integer.

Lesson 9-6 The Complex Plane and Polar Form of Complex Numbers 587

If z � a � bi, then z � �a2 ��b2�.Absolute Value

of a ComplexNumber

i

�

(3, 2)

O

i

�

(0, 4)

O

i

b

a

r

�

�O

Examples 2Re

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Example 3

i

�

z2z1

z4

z3

z51

1

�1

�1O

As with other rectangular coordinates, complex coordinates can be written inpolar form by substituting a � r cos � and b � r sin �.

z � a � bi� r cos � � (r sin �)i� r (cos � � i sin �)

This form of a complex number is often called the polar or trigonometricform.

Values for r and � can be found by using the same process you used when changing rectangular coordinates to polar coordinates. For a � bi, r � �a2 � b�2�and � � Arctan �

ab

� if a � 0 or � � Arctan �ab

� � � if a � 0. The amplitude � is usually expressed in radian measure, and the angle is in standard position along thepolar axis.

Express each complex number in polar form.

a. �3 � 4i

First, plot the number in the complex plane.

Then find the modulus.

r � �(�3)2� � 42� or 5

Now find the amplitude. Notice that � is in Quadrant II.

� � Arctan ��

43� � �

� 2.21

Therefore, �3 � 4i � 5(cos 2.21 � i sin 2.21) or 5 cis 2.21.

b. 1 � �3�i

First, plot the number in the complex plane.

Then find the modulus.

r � �12 � ��3��2� or 2

Now find the amplitude. Notice that � is inQuadrant I.

� � Arctan or ��

3�

Therefore, 1 ��3�i � 2 �cos ��

3� � i sin �

�

3�� or 2 cis �

�

3�.

�3��1


The polar form or trigonometric form of the complex number a � bi is

r (cos � � i sin �).

Polar Form ofa Complex

Number

r (cos � � i sin �) is often abbreviated as r cis �.

i

�

(�3, 4)

�

O

i

�O 1�1�1

�2

�2 2

2

1

�

1,� 3)(

Example 4


Guided Practice

You can also graph complex numbers in polar form.

Graph 4�cos �11

6�� i sin �

116

��. Then express it in rectangular form.

In the polar form of this complexnumber, the value of r is 4, and the

value of � is �11

6��. Plot the point

with polar coordinates �4, �11

6��.

To express the number inrectangular form, simplify thetrigonometric values:

4�cos �11

6�� i sin �

116��

� 4� � i��12

�� 2�3� � 2i

�3��2


O0

125�

127�

1223�

611�

1219�

1217�

�

1213�

1211�

12�

2�

6�

3�

4�

23�

47�

34�

45�

65�

43�

32�

67�

35�

1 2 3 4

i

�


1. Explain how to find the absolute value of a complex number.

2. Write the polar form of i.

3. Find a counterexample to the statement z1 � z2 � z1 � z2 for all complexnumbers z1 and z2.

4. Math Journal Your friend is studying complex numbers at another school atthe same time that you are. She learned that the absolute value of a complexnumber is the square root of the product of the number and its conjugate. Youknow that this is not how you learned it. Write a letter to your friend explainingwhy this method gives the same answer as the method you know. Use algebra,but also include some numerical examples of both techniques.

5. Solve the equation 2x � y � xi � yi � 5 � 4i for x and y, where x and y are real numbers.


6. �2 � i 7. 1 � �2�i


8. 2 � 2i 9. 4 � 5i 10. �2

Graph each complex number. Then express it in rectangular form.

11. 4�cos ��

3� � i sin �

�

3�� 12. 2(cos 3 � i sin 3) 13. �

32

�(cos 2� � i sin 2�)

14. Graph the first five members of the orbit of z0 � �0.25 � 0.75i under iterationby f(z) � z2 � 0.5.


Example 5

Practice


15. Vectors The force on an object is represented by the complex number 10 � 15i, where the components are measured in newtons.a. What is the magnitude of the force?b. What is the direction of the force?

Solve each equation for x and y, where x and y are real numbers.

16. 2x � 5yi � 12 � 15i 17. 1 � (x � y)i � y � 3xi

18. 4x � yi � 5i � 2x � y � xi � 7i


19. 2 � 3i 20. 3 � 4i 21. �1 � 5i

22. �3i 23. �1 � �5�i 24. 4 � �2�i

25. Find the modulus of z � �4 � 6i.


26. 3 � 3i 27. �1 � �3�i 28. 6 � 8i

29. �4 � i 30. 20 � 21i 31. �2 � 4i

32. 3 33. �4�2� 34. �2i

Graph each complex number. Then express it in rectangular form.

35. 3�cos ��

4� � i sin �

�

4�� 36. cos ��

�

6�� i sin ��

�

6��

37. 2�cos �43�� i sin �

43�� 38. 10(cos 6 � i sin 6)

39. 2�cos �54�� i sin �

54�� 40. 2.5(cos 1 � i sin 1)

41. 5(cos 0 � i sin 0) 42. 3(cos � � i sin �)

Graph the first five members of the orbit of each initial value under iteration bythe given function.

43. z0 � �0.5 � i, f(z) � z2 � 0.5 44. z0 � � i, f(z) � z2

45. Graph the first five iterates of z0 � 0.5 � 0.5i under f(z) � z2 � 0.5.

46. Electrical Engineering Refer to Exercise 44 in Lesson 9-3. Consider a circuit with alternating current that contains two voltage sources in series.Suppose these two voltages are given by v1(t) � 40 sin (250t � 30°) and v2(t) � 60 sin (250t � 60°), where t represents time, in seconds.a. The phasors for these two voltage sources are written as 40�30°

and 60�60°, respectively. Convert these phasors to complex numbers in rectangular form. (Use j as the imaginary unit, as electrical engineers do.)

b. Add these two complex numbers to find the total voltage in the circuit.c. Write a sinusoidal function that gives the total voltage in the circuit.

�2��2

�2��2


E XERCISES

A

B

C

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Mixed Review

47. Critical Thinking How are the polar forms of complex conjugates alike? Howare they different?

48. Electricity A series circuit contains two sources of impedance, one of 10(cos 0.7 � j sin 0.7) ohms and the other of 16(cos 0.5 � j sin 0.5) ohms.

a. Convert these complex numbers to rectangular form.

b. Add your answers from part a to find the total impedance in the circuit.

c. Convert the total impedance back to polar form.

49. Transformations Certain operations with complex numbers correspond togeometric transformations in the complex plane. Describe the transformationapplied to point z to obtain point w in the complex plane for each of thefollowing operations.

a. w � z � (2 � 3i)

b. w � i � z

c. w � 3z

d. w is the conjugate of z

50. Critical Thinking Choose any two complex numbers, z1 and z2, inrectangular form.

a. Find the product z1z2.

b. Write z1, z2, and z1z2 in polar form.

c. Repeat this procedure with a different pair of complex numbers.

d. Make a conjecture about the product of two complex numbers in polarform.

51. Simplify (6 � 2i)(�2 � 3i). (Lesson 9-5)

52. Find the rectangular coordinates of the point with polar coordinates (�3,�135°). (Lesson 9-3)

53. Find the magnitude of the vector ��3,7�, and write the vector as a sum of unitvectors. (Lesson 8-2)

54. Use a sum or difference identity to find tan 105°. (Lesson 7-3)

55. Mechanics A pulley of radius 18 centimeters turns at 12 revolutions persecond. What is the linear velocity of the belt driving the pulley in meters persecond? (Lesson 6-2)

56. If a � 12 and c � 18 in �ABC, find the measure ofangle A to the nearest tenth of a degree. (Lesson 5-5)

57. Solve �2a � 1� � �3a � 5�. (Lesson 4-7)

58. Without graphing, describe the end behavior of the graph of y � 2x2 � 2. (Lesson 3-5)

59. SAT/ACT Practice A person is hired for a job that pays $500 per month andreceives a 10% raise in each following month. In the fourth month, how muchwill that person earn?

A $550 B $600.50 C $650.50 D $665.50 E $700


c a

bA

B

C


9-6B Geometry in theComplex PlaneAn Extension of Lesson 9-6

Many geometric figures and relationships can be described by using complexnumbers. To show points on figures, you can store the real and imaginary partsof the complex numbers that correspond to the points in lists L1 and L2 and useSTAT PLOT to graph the points.

1. Store �1 � 2i as M and 1 � 5i as N. Now consider complex numbers of theform (1 � T)M � TN, where T is a real number. You can generate severalnumbers of this form and store their real and imaginary parts in L1 and L2,respectively, by entering the following instructions on the home screen.

seq( is in the LIST OPS menu. real( and imag( are in the MATH CPX menu.

Use a graphing window of [�10, 10] sc1:1 by [�25, 25] sc1:5. Turn on Plot 1 anduse a scatter plot to display the points defined in L1 and L2. What do you noticeabout the points in the scatter plot?

2. Are the original numbers M and N shown in the scatter plot? Explain.

3. Repeat Exercise 1 storing �1 � 1.5i as M and �2 � i as N. Describe yourresults.

4. Repeat Exercises 1 and 2 for several complex numbers M and N of your choice. (You may need to change the window settings.) Then make aconjecture about where points of the form (1 � T)M � TN are located inrelation to M and N.

5. Suppose K, M, and N are three noncollinear points in the complex plane. Where will you find all the points that can be expressed in the form aK � bM � cN, where a, b, and c are nonnegative real numbers such that a � b � c � 1? Use the calculator to check your answer.

6. In Exercises 1-4, where is (1 � T)M � TN in relation to M and N if the value of T is between 0 and 1?

7. Where in the complex plane will you find the complex numbers z that satisfythe equation z � (1 � i) � 5?

8. What equation models the points in the complex plane that lie on the circle ofradius 2 that is centered at the point 2 � 3i?


TRY THESE

WHAT DO YOUTHINK?

OBJECTIVE• Explore

geometricrelationships inthe complexplane.

GRAPHING CALCULATOR EXPLORATION

Lesson 9-7 Products and Quotients of Complex Numbers in Polar Form 593

Products and Quotients of Complex Numbers in Polar Form

ELECTRICITY Complex numbers can be used in the study of electricity,specifically alternating current (AC). There are three basic quantities toconsider:• the current I, measured in amperes,• the impedance Z to the current, measured in ohms, and• the electromotive force E or voltage, measured in volts.

These three quantities are related by the equation E � I � Z. Current, impedance, andvoltage can be expressed as complex numbers. Electrical engineers use j as theimaginary unit, so they write complex numbers in the form a � bj. For the totalimpedance a � bj, the real part arepresents the opposition to current flow due to resistors, and the imaginary part b is related to the opposition due to inductors and capacitors. If a circuit has a total impedance of 2 � 6j ohms and a voltage of 120 volts, find the current in the circuit. This problem will be solved in Example 3.

Multiplication and division of complex numbers in polar form are closely tiedto geometric transformations in the complex plane. Let r1(cos �1 � i sin �1) andr2(cos �2 � i sin �2) be two complex numbers in polar form. A formula for theproduct of the two numbers can be derived by multiplying the two numbersdirectly and simplifying the result.

r1(cos �1 � i sin �1) � r2(cos �2 � i sin �2)

� r1r2(cos �1 cos �2 � i cos �1 sin �2 � i sin �1 cos �2 � i2 sin �1 sin �2)

� r1r2[(cos �1 cos �2 � sin �1 sin �2) � i(sin �1 cos �2 � cos �1 sin �2)] i2 � �1

� r1r2[cos (�1 � �2) � i sin (�1 � �2)] Sum identities for cosine and sine

9-7

Real World

Ap

plic ation

OBJECTIVE• Find the product

and quotient of complexnumbers inpolar form.

r1(cos �1 � i sin �1) � r2(cos �2 � i sin �2) �r1r2[cos (�1 � �2) � i sin (�1 � �2)]

Product ofComplex

Numbers inPolar Form

Notice that the modulus (r1r2) of the product of the two complex numbers isthe product of their moduli. The amplitude (�1 � �2) of the product is the sum ofthe amplitudes.

Find the product 3�cos �76�� i sin �

76�� 2�cos �

23�� i sin �

23��. Then express

the product in rectangular form.

Find the modulus and amplitude of the product.

r � r1r2 � � �1 � �2

� 3(2) � �76��

23��

� 6 � �11

6��

The product is 6�cos �11

6�� i sin �

116��.

Now find the rectangular form of the product.

6�cos �11

6�� i sin �

116�� 6��

�2

3��

12

� i� cos �11

6��

�2

3��, sin �

116��

12

�

� 3�3� � 3i

The rectangular form of the product is 3�3� � 3i.

Suppose the quotient of two complex numbers is expressed as a fraction. Aformula for this quotient can be derived by rationalizing the denominator. Torationalize the denominator, multiply both the numerator and denominator bythe same value so that the resulting new denominator does not contain imaginarynumbers.

� �

� �

� [cos (�1 � �2) � i sin (�1 � �2)] Trigonometric identities

Notice that the modulus ��rr1

2�� of the quotient of two complex numbers is the

quotient of their moduli. The amplitude (�1 � �2) of the quotient is the differenceof the amplitudes.

r1�r2

(cos �1 cos �2 � sin �1 sin �2) � i(sin �1 cos �2 � cos �1 sin �2)��

cos2 �2 � sin2 �2

r1�r2

(cos �2 � i sin �2)��(cos �2 � i sin �2)

r1(cos �1 � i sin �1)��r2(cos �2 � i sin �2)



Example 1

cos �2 � i sin �2 is theconjugate of cos �2 � i sin �2.

� �rr1

2� [cos (�1 � �2) � i sin (�1 � �2)]


Quotient ofComplex

Numbers inPolar Form

Find the quotient 12�cos ��4

� � i sin ��4

�� 4�cos �32�� i sin �

32��. Then express

the quotient in rectangular form.

Find the modulus and amplitude of the quotient.

r � � � �1 � �2

� �142� � �

�

4� � �

32��

� 3 � � �54��

The quotient is 3�cos ��54�� i sin ��

54��.

Now find the rectangular form of the quotient.

3�cos ��54�� i sin ��

54�� 3�� i�

� � � i

The rectangular form of the quotient is � � i.

You can use products and quotients of complex numbers in polar form tosolve the problem presented at the beginning of the lesson.

ELECTRICITY If a circuit has an impedance of 2 � 6j ohms and a voltage of120 volts, find the current in the circuit.


120 � 120(cos 0 � j sin 0)2 � 6j � �40�[cos (�1.25) � j sin (�1.25)]

� 2�10�[cos (�1.25) � j sin (�1.25)]

Substitute the voltage and impedance into the equation E � I � Z.

E � I � Z

120(cos 0 � j sin 0) � I � 2�10�[cos (�1.25) � j sin (�1.25)]

� I

6�10�(cos 1.25 � j sin 1.25) � I

Now express the current in rectangular form.

I � 6�10�(cos 1.25 � j sin 1.25)

� 5.98 � 18.01j

The current is about 6 � 18j amps.

120(cos 0 � j sin 0)��2�10�[cos (�1.25) � j sin (�1.25)]

3�2��2

3�2��2

3�2��2

3�2��2

�2��2

�2��2

r1�r2


cos ��54�� ,

sin ��54�� 2��

2

�2��2

Example 2

Example 3

Real World

Ap

plic ation

Use a calculator. 6�10� cos 1.25 � 5.98,

6�10� sin 1.25 � 18.01

r � �22 � (��6)2� � �40� or 2�10�,

� � Arctan ��26� or �1.25


Guided Practice

Practice



1. Explain how to find the quotient of two complex numbers in polar form.

2. Describe how to square a complex number in polar form.

3. List which operations with complex numbers you think are easier in rectangularform and which you think are easier in polar form. Defend your choices withexamples.

Find each product or quotient. Express the result in rectangular form.

4. 2�cos ��

2� � i sin �

�

2�� 2�cos �

32�� i sin �

32��

5. 3�cos ��

6� � i sin �

�

6�� 4�cos �

23�� i sin �

23��

6. 4�cos �94�� i sin �

94�� 2�cos ��

�

2�� i sin ��

�

2��

7. �12

��cos ��

3� � i sin �

�

3�� 6�cos �

56�� i sin �

56��

8. Use polar form to find the product �2 � 2�3�i� � ��3 � �3�i�. Express theresult in rectangular form.

9. Electricity Determine the voltage in a circuit when there is a current of

2�cos �11

6�� j sin �

116�� amps and an impedance of 3�cos �

�

3� � j sin �

�

3�� ohms.

Find each product or quotient. Express the result in rectangular form.

10. 4�cos ��

3� � i sin �

�

3�� 7�cos �

23�� i sin �

23��

11. 6�cos �34�� i sin �

34�� 2�cos �

�

4� � i sin �

�

4��

12. �12

��cos ��

3� � i sin �

�

3�� 3�cos �

�

6� � i sin �

�

6��

13. 5(cos � � i sin �) � 2�cos �34�� i sin �

34��

14. 6�cos ��

3�� i sin ��

�

3�� 3�cos �

56�� i sin �

56��

15. 3�cos �73�� i sin �

73�� cos �

�

2� � i sin �

�

2��

16. 2(cos 240° � i sin 240°) � 3(cos 60° � i sin 60°)

17. �2��cos �74�� i sin �

74�� cos �

34�� i sin �

34��

18. 3(cos 4 � i sin 4) � 0.5(cos 2.5 � i sin 2.5)

�2��

2


E XERCISES

A

B




Mixed Review

19. 4[cos (�2) � i sin (�2)] � (cos 3.6 � i sin 3.6)

20. 20�cos �76�� i sin �

76�� 15�cos �

113�� i sin �

113��

21. 2�cos �34�� i sin �

34�� 2��cos �

�

2� � i sin �

�

2��

22. Find the product of 2�cos ��

3� � i sin �

�

3�� and 6�cos ��

�

6�� i sin ��

�

6��. Write the

answer in rectangular form.

23. If z1 � 4�cos �53�� i sin �

53�� and z2 � �

12

��cos ��

3� � i sin �

�

3��, find �

zz

1

2� and express

the result in rectangular form.

Use polar form to find each product or quotient. Express the result inrectangular form.

24. (2 � 2i) � (�3 � 3i) 25. ��2� � �2�i� � ��3�2� � 3�2�i�26. ��3� � i� � �2 � 2�3�i� 27. ��4�2� � 4�2�i� � (6 � 6i)

28. Electricity Find the current in a circuit with a voltage of 13 volts and animpedance of 3 � 2j ohms.

29. Electricity Find the impedance in a circuit with a voltage of 100 volts and acurrent of 4 � 3j amps.

30. Critical Thinking Given z1 and z2graphed at the right, graph z1z2 and

�zz

1

2� without actually calculating them.

31. Transformationsa. Describe the transformation

applied to the graph of thecomplex number z if z ismultiplied by cos � � i sin �.

b. Describe the transformationapplied to the graph of thecomplex number z if z is

multiplied by �12

� � i.

32. Critical Thinking Find the quadratic equation az2 � bz � c � 0 such that a � 1

and the solutions are 3�cos ��

3� � i sin �

�

3�� and 2�cos �

56�� i sin �

56��.

33. Express 5 � 12i in polar form. (Lesson 9-6)

34. Write the equation r � 5 sec�� 56�� in rectangular form. (Lesson 9-4)

35. Physics A prop for a play is supported equally by two wires suspended fromthe ceiling. The wires form a 130° angle with each other. If the prop weighs 23 pounds, what is the tension in each of the wires? (Lesson 8-5)

�3��

2


O0

125�

127�

1223�

611�

1219�

1217�

�

1213�

1211�

12�

2�

6�

3�

4�

23�

47�

34�

45�

65�

43�

32�

67�

35�

1 2

z2

z1

3 4

i

�

C

Real World

Ap

plic ation


36. Solve cos 2x � sin x � 1 for principal values of x. (Lesson 7-5)

37. Write the equation for the inverse of y � cos x. (Lesson 6-8)

38. SAT/ACT Practice In the figure, the perimeter of squareBCDE is how much smaller than the perimeter ofrectangle ACDF?

A 2 B 3 C 4

D 6 E 16

2

3

A

F D

C

E

B

Have you ever gazedinto the sky at night

hoping to spot aconstellation? Doyou dream ofhaving your owntelescope? If you

enjoy studyingabout the universe,

then a career inastronomy may be just for

you. Astronomers collect and analyze data about the universe including stars, planets,comets, asteroids, and even artificialsatellites. As an astronomer, you may collect information by using a telescope orspectrometer here on earth, or you may useinformation collected by spacecraft andsatellites.

Most astronomers specialize in onebranch of astronomy such as astrophysicsor celestial mechanics. Astronomers oftenteach in addition to conducting research.Astronomers located throughout the worldare prime sources of information for NASAand other countries’ space programs.

CAREER OVERVIEW

Degree Preferred:at least a bachelor’s degree in astronomy orphysics

Related Courses:mathematics, physics, chemistry, computerscience

Outlook:average through the year 2006

Astronomer

For more information on careers in astronomy, visit: www.amc.glencoe.com

CAREER CHOICES


Space Program Spending

Current Year DollarValue

‘600 ‘64 ‘68 ‘72 ‘76 ‘80 ‘84 ‘88 ‘92 ‘96

25,000

20,000

15,000

10,000

5,000

1996 Dollar Value

Dollars (millions)

YearSource: National Aeronautics and Space Administration


Lesson 9-8 Powers and Roots of Complex Numbers 599

Powers and Roots of Complex Numbers

COMPUTER GRAPHICS Many of the computer graphics that arereferred to as fractals are graphs of Julia sets, which are named afterthe mathematician Gaston Julia. When a function like f(z) � z2 � c,

where c is a complex constant, is iterated, points in the complex plane can beclassified according to their behavior under iteration.

• Points that escape to infinity under iteration belong to the escape set of the function.

• Points that do not escape belong to the prisoner set.

The Julia set is the boundary between the escape set and the prisoner set. Is the number w � 0.6 � 0.5i in the escape set or the prisoner set of the function f(z) � z2? This problem will be solved in Example 6.

You can use the formula for the product of complex numbers to find thesquare of a complex number.

[r(cos � � i sin �)]2 � [r(cos � � i sin �)] � [r(cos � � i sin �)]

� r2[cos (� � �) � i sin (� � �)]

� r2(cos 2� � i sin 2�)

Other powers of complex numbers can be found using De Moivre’s Theorem.

Find �2 � 2�3�i�6.

First, write 2 � 2�3�i in polar form. Note that its graph is in the first quadrantof the complex plane.

r � �22 � ��2�3��2� � � Arctan

� �4 � 12� � Arctan �3�� 4 � �

�

3�

2�3��

2

9-8

Real World

Ap

plic ation

OBJECTIVE• Find powers

and roots ofcomplexnumbers inpolar form usingDe Moivre’sTheorem.

[r (cos � � i sin �)]n � rn(cos n� � i sin n�)De Moivre’sTheorem

You will be asked to prove De Moivre’s Theorem in Chapter 12.

Example 1


The polar form of 2 � 2�3�i is 4�cos ��

3� � i sin �

�

3��.

Now use De Moivre’s Theorem to find the sixth power.

(2 � 2�3�i)6 � �4�cos ��

3� � i sin �

�

3��6

� 46�cos 6��

3�� i sin 6��

�

3��

� 4096(cos 2� � i sin 2�)

� 4096(1 � 0i) Write the result in rectangular form.

� 4096

Therefore, �2 � 2�3�i�6� 4096.

De Moivre’s Theorem is valid for all rational values of n. Therefore, it is alsouseful for finding negative powers of complex numbers and roots of complexnumbers.

Find � � �12

�i��5

.

First, write � �12

� i in polar form. Note that its graph is in the fourth

quadrant of the complex plane.

r � ��2

3��2

� ��12

��2� � � Arctan

� �34

� � �14

�� or 1 � Arctan �� or ��

6�

The polar form of � �12

�i is 1�cos ��

6�� i sin ��

�

6��.

Use De Moivre’s Theorem to find the negative 5th power.

� � �12

�i��5� �1�cos ��

�

6�� i sin ��

�

6��5

� 1�5�cos (�5)��

6�� i sin (�5)��

�

6�� De Moivre’s Theorem

� 1�cos �56�� i sin �

56�� Simplify.

� � � �12

�i Write the answer in rectangular form.�3��

2

�3��

2

�3��

2

�3��

3

��12

��

��2

3��

�3��

2

�3��

2


Example 2

Recall that positive real numbers have two square roots and that the positiveone is called the principal square root. In general, all nonzero complex numbershave p distinct pth roots. That is, they each have two square roots, three cuberoots, four fourth roots, and so on. The principal pth root of a complex number isgiven by:

(a � bi)�p1

�

� [r(cos � � i sin �)]�p1

�

� r�p1

��cos �p�

� � i sin �p�

��.

Find �38i�.

�38i� � (0 � 8i)

�13

�a � 0, b � 8

Polar form; r � �02 � 8�2� or 8, � � ��2

�� 8�cos �

�

2� � i sin �

�

2��

�13

�

since a � 0.

� 8�13

��cos ��13

��

2�� i sin ��

13

��

2�� De Moivre’s Theorem

� 2�cos ��

6� � i sin �

�

6��

� 2��2

3��

12

�i� or �3� � i This is the principal cube root.

The following formula generates all of the pth roots of a complex number. It isbased on the identities cos � � cos (� � 2n�) and sin � � sin (� � 2n�), where nis any integer.

Find the three cube roots of �2 � 2i.

First, write �2 � 2i in polar form.

r � �(�2)2� � (��2)2� or 2�2� � � Arctan ��

�

22� � � or �

54��

�2 �2i � 2�2��cos ��54�� 2n�� i sin ��

54�� 2n�� n is any integer.

Now write an expression for the cube roots.

(�2 � 2i)�13

� � �2�2��cos ��54�� 2n�� i sin ��

54�� 2n��

�13

�

� �2��cos � � � i sin � ��(continued on the next page)

�54�� 2n�

��3

�54�� 2n�

��3


Example 3

The p distinct pth roots of a � bi can be found by replacing n with 0, 1, 2, …, p � 1, successively, in the following equation.

(a � bi )�1p�

� (r [cos (� � 2n�) � i sin (� � 2n�)])�1p�

� r�1p��cos ��

p2n�� i sin ��

p2n��

The p Distinctpth Roots of

a ComplexNumber

Example 4

When finding a principal root, the interval�� is used.

�2�2��13

�

� �2�32

��13

�

� 2�12

�or �2�

Let n � 0, 1, and 2 successively to find the cube roots.

Let n � 0. �2��cos � � � i sin � �� 2��cos �

51�

2� � i sin �

51�

2��

� 0.37 � 1.37i

Let n � 1. �2��cos � � � i sin � �� 2��cos �

1132�� i sin �

1132��

� �1.37 � 0.37i

Let n � 2. �2��cos � � � i sin � �� 2��cos �

2112�� i sin �

2112��

� 1 � i

The cube roots of �2 � 2i are approximately 0.37 � 1.37i, �1.37 � 0.37i, and 1 � i. These roots can be checked by multiplication.

�54�� 2(2)�

��3

�54�� 2(2)�

��3

�54�� 2(1)�

��3

�54�� 2(1)�

��3

�54�� 2(0)�

��3

�54�� 2(0)�

��3

GRAPHING CALCULATOR EXPLORATION

The p distinct pth roots of a complex numbercan be approximated using the parametricmode on a graphing calculator. For a particularcomplex number r(cos � � i sin �) and aparticular value of p:

➧ Select the Radian and Par modes.

➧ Select the viewing window.

Tmin � �p�

�, Tmax � �p�

� � 2�, Tstep � �2p��,

Xmin � �r�p1

�

, Xmax � r�p1

�

, Xscl � 1,

Ymin � �r�p1

�

, Ymax � r�p1

�

, and Yscl � 1.

➧ Enter the parametric equations

X1T � r�p1

�

cos T and Y1T � r�p1

�

sin T.

➧ Graph the equations.

➧ Use to locate the roots.

TRY THESE

1. Approximate the cube roots of 1.

2. Approximate the fourth roots of i.

3. Approximate the fifth roots of 1 � i.

WHAT DO YOU THINK?

4. What geometric figure is formed when yougraph the three cube roots of a complexnumber?

5. What geometric figure is formed when yougraph the fifth roots of a complex number?

6. Under what conditions will the complexnumber a � bi have a root that lies on thepositive real axis?

TRACE


You can also use De Moivre’s Theorem to solve some polynomial equations.

Solve x5 � 32 � 0. Then graph the roots in the complex plane.

The solutions to this equation are the same as those of the equation x5 � 32.That means we have to find the fifth roots of 32.

32 � 32 � 0i a � 32, b � 0

� 32(cos 0 � i sin 0) Polar form; r � �322 �� 02� or 32, � � Arctan �302� or 0

Now write an expression for the fifth roots.

32�15

� � [32(cos (0 � 2n�) � i sin (0 � 2n�))]�15

�

� 2�cos �2n

5�� i sin �

2n5��

Let n � 0, 1, 2, 3, and 4 successively to find the fifth roots, x1, x2, x3, x4, and x5.

Let n � 0. x1 � 2(cos 0 � i sin 0) � 2

Let n � 1. x2 � 2�cos �25�� i sin �

25�� 0.62 � 1.90i

Let n � 2. x3 � 2�cos �45�� i sin �

45�� 1.62 � 1.18i

Let n � 3. x4 � 2�cos �65�� i sin �

65�� 1.62 � 1.18i

Let n � 4. x5 � 2�cos �85�� i sin �

85�� 0.62 � 1.90i

The solutions of x5 � 32 � 0 are 2, 0.62 1.90i, and �1.62 1.18i.

The solutions are graphed at the right. Noticethat the points are the vertices of a regularpentagon. The roots of a complex number arecyclical in nature. That means, when theroots are graphed on the complex plane, theroots are equally spaced around a circle.

COMPUTER GRAPHICS Refer to the application at the beginning of thelesson. Is the number w � 0.6 � 0.5i in the escape set or the prisoner set ofthe function f(z) � z2?

Iterating this function requires you to square complex numbers, so you canuse De Moivre’s Theorem.

Write w in polar form. r � �0.62 �� (�0.5�)2� or about 0.78

w � 0.78[cos (�0.69) � i sin (�0.69)] � � Arctan ��

00.6.5� or about �0.69


i

�

�1.62 � 1.18i

�1.62 � 1.18i

0.62 � 1.90i

0.62 � 1.90i

2O�2

�2

�1

�1

1

1

2

Examples 5

Real World

Ap

plic ation

Example 6



Now iterate the function.w1 � f(w)

� w2

� (0.78[cos (�0.69) � i sin (�0.69)])2

� 0.782[cos 2(�0.69) � i sin 2(�0.69)] De Moivre’s Theorem

� 0.12 � 0.60i Use a calculator to approximatethe rectangular form.

w2 � f(w1)

� w12

� (0.782[cos 2(�0.69) � i sin 2(�0.69)])2 Use the polar form of w1.

� 0.784[cos 4(�0.69) � i sin 4(�0.69)]

� �0.34 � 0.14i

w3 � f(w2)

� w22

� (0.784[cos 4(�0.69) � i sin 4(�0.69)])2 Use the polar form of w2.

� 0.788[cos 8(�0.69) � i sin 8(�0.69)]

� 0.10 � 0.09i

The moduli of these iterates are 0.782, 0.784, 0.788, and so on. These moduli will approach 0 as the number of iterations increases. This means the graphs of the iterates approach the origin in the complex plane, so w � 0.6 � 0.5i is in the prisoner set of the function.


GraphingCalculatorProgramsFor a programthat drawsJulia sets, visit:www.amc.glencoe.com

w

i

�O

1

�1

1�1 w2

w3

w1


1. Evaluate the product (1 � i) (1 � i) (1 � i) (1 � i) (1 � i) by traditionalmultiplication. Compare the results with the results using De Moivre’s Theoremon (1 � i)5. Which method do you prefer?

2. Explain how to use De Moivre’s Theorem to find the reciprocal of a complexnumber in polar form.

3. Graph all the fourth roots of a complex number if a � ai is one of the fourthroots. Assume a is positive.

4. You Decide Shembala says that if a � 0, then (a � ai)2 must be a pureimaginary number. Arturo disagrees. Who is correct? Use polar form to explain.



Guided Practice

Practice

GraphingCalculator


Find each power. Express the result in rectangular form.

5. ��3� � i�36. (3 � 5i)4

Find each principal root. Express the result in the form a � bi with a and brounded to the nearest hundredth.

7. i�16

�

8. (�2 � i)�13

�

Solve each equation. Then graph the roots in the complex plane.

9. x4 � i � 0 10. 2x3 � 4 � 2i � 0

11. Fractals Refer to the application at the beginning of the lesson. Is w � 0.8 � 0.7i in the prisoner set or the escape set for the function f(z) � z2?Explain.

Find each power. Express the result in rectangular form.

12. �3�cos ��

6� � i sin �

�

6��3

13. �2�cos ��

4� � i sin �

�

4��5

14. (�2 � 2i)3 15. �1 � �3�i�4

16. (3 � 6i )4 17. (2 � 3i)�2

18. Raise 2 � 4i to the fourth power.

Find each principal root. Express the result in the form a � bi with a and brounded to the nearest hundredth.

19. �32�cos �23�� i sin �

23��

�15

�

20. (�1)�14

�

21. (�2 � i)�14

�22. (4 � i)

�13

�

23. (2 � 2i)�13

�24. (�1 � i)

�14

�

25. Find the principal square root of i.

Solve each equation. Then graph the roots in the complex plane.

26. x3 � 1 � 0 27. x5 � 1 � 0

28. 2x4 � 128 � 0 29. 3x4 � 48 � 0

30. x4 � (1 � i ) � 0 31. 2x4 � 2 � 2�3�i � 0

Use a graphing calculator to find all of the indicated roots.

32. fifth roots of 10 � 9i 33. sixth roots of 2 � 4i

34. eighth roots of 36 � 20i

35. Fractals Is the number �12

� � �34

� i in the escape set or the prisoner set for the function f(z) � z2? Explain.

36. Critical Thinking Suppose w � a � bi is one of the 31st roots of 1.a. What is the maximum value of a?b. What is the maximum value of b?


E XERCISES

A

B

C

Real World

Ap

plic ation



Mixed Review

37. Design Gloribel works for an advertising agency. She wants to incorporate ahexagon design into the artwork for one of her proposals. She knows that shecan locate the vertices of a regular hexagon by graphing the solutions to theequation x6 � 1 � 0 in the complex plane. What are the solutions to thisequation?

38. Computer Graphics Computer programmers can use complex numbers andthe complex plane to implement geometric transformations. If a programmerstarts with a square with vertices at (2, 2), (�2, 2), (�2, �2), and (2, �2), eachof the vertices can be stored as a complex number in polar form. Complexnumber multiplication can be used to rotate the square 45° counterclockwiseand dilate it so that the new vertices lie at the midpoints of the sides of theoriginal square.a. What complex number should the programmer multiply by to produce this

transformation?b. What happens if the original vertices are multiplied by the square of your

answer to part a?

39. Critical Thinking Explain why the sum of the imaginary parts of the p distinctpth roots of any positive real number must be zero.

40. Find the product 2�cos ��

6� � i sin �

�

6�� 3�cos �

53�� i sin �

53��. Express the result in

rectangular form. (Lesson 9-7)

41. Simplify (2 � 5i) � (�3 � 6i) � (�6 � 2i). (Lesson 9-5)

42. Write parametric equations of the line with equation y � �2x � 7. (Lesson 8-6)

43. Use a half-angle identity to find the exact value of cos 22.5°. (Lesson 7-4)

44. Solve triangle ABC if A � 81°15 and b � 28. Roundangle measures to the nearest minute and sidemeasures to the nearest tenth. (Lesson 5-4)

45. Manufacturing The PreciousAnimal Company must produceat least 300 large stuffed bearsand 400 small stuffed bears perday. At most, the company canproduce a total of 1200 bears per day. The profit for each large bear is $9.00, and the profit for each small bear is$5.00. How many of each type of bear should be produced each day to maximize profit?(Lesson 2-7)

46. SAT/ACT Practice Six quarts of a 20% solution of alcohol in water are mixedwith 4 quarts of a 60% solution of alcohol in water. The alcoholic strength of themixture is

A 36% B 40% C 48% D 60% E 80%


c a

bA

B

C


9-5 simplifying complex numberscentros.edu.xunta.es/.../dptos/mat/jr/varios/complex_numbers.pdf ·...

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